I've got to be honest this arrangement sounds like more trouble than it's worth.
Your mil will either give what she wants and lie about it or give what she wants and tell you she has.
I don't think talking to her will do anything but inflame the situation.
I think you either need to make peace with the fact she will do it her way or not leave you dc with her.
If your dp says if she doesn't go to his mum then she doesn't go to yours then I'd call his bluff and tell him that you're going to use nursery full time.
You need to be honest with yourself do you genuinely think she will put your child in danger while caring for them either emotionally, negligently or physically or do you resent having to leave your dc with her?
I have an exdh and know how that feels even if we aren't supposed to feel that way in mumsnet land
If it's the first that you genuinely believe she will put your dc in danger then you need to raise this with your dp and make it clear your not prepared to risk dc in order to pander to his mum.
I do not trust my pil with my dc and have made it clear to DH they will not be caring for them alone. I was able to justify it to him by giving examples of behaviours they did that concerned me. For example they think dc (2) should be left alone to play on the streets, they think dc (2) can be left alone in the bath while they're downstairs so long as they can hear him 'splashing every so often'. I know this because they have suggested all of these things to me much to my horror. When I presented them to my DH and asked if he thought this was a safe way of our dc being cared for he was kinda stumped because he couldn't agree despite the emotional blackmail from his parents.
But downside is I don't rely on my parents for regular childcare too as it would just cause untold drama for DH. Although we do visit a lot more than we ever let on to pil and leave the dc with my parents for nights out and nipping out type thing. We would never tell pil this.
However if it's because you can't stand her and hate the thought of leaving you dc with her then you either need to find a way to get past it and agree or not let either grandmother provide care for your dd.
If she's only having her one day a week then even if she does give her a chocolate bar or Nutella on toast it's really not going to harm her.
On the occasions I've needed emergency childcare for ds and he's gone to my dad he eats a lot of crap I wouldn't give him at home but I know actually won't do him any harm one day a week.
The truth is you'll never find anyone who will care for dc the exact way you want them to even your own mum even your dp because no two people parent the same but so long as they're safe and loved by people who cares for them that's all that matters